Some people argue that it is every Christian's duty to do the work of an evangelist. I'm not sure. The distinctive work of an evangelist is to proclaim the gospel. Preaching and proclamation comprise one of the offices in the New Testament and one of the gifts of the Spirit, and so on. This gift is not given to everybody, and so I would say that, in the technical sense, it is not every Christian's responsibility to be an evangelist.
The New Testament does make it clear that every Christian is to be a witness. Some of the confusion enters at this point because in Christian jargon "witness" is often used as a synonym for "evangelist." The New Testament makes a clear distinction between the two words, a distinction between the general and the particular. "Witness" is the broad statement. To bear witness to something is to make something visible that is not readily visible, or not manifest, but invisible. The New Testament word for "witness" is martyria, which is the word from which we get the English wordmartyr. Those who died for the faith bore witness, or made manifest, their commitment to Christ. That was one way of doing it, but it wasn't the work of evangelism.
Evangelism is one specific form of witnessing. Every Christian is called to witness; every Christian is called to confess Jesus with their speech as well as their actions. God doesn't ordain us to be secret service Christians. But not everyone is called, in my judgment, to be an evangelist; that's a special task. I think every Christian has a responsibility to participate in the evangelistic enterprise. While not everybody is called to be a missionary, we are all called to do our part in the mission of the church. The church is given the responsibility of the great commission, and every member in the body of Christ is called to do his or her part in seeing to it that the task is done. Evangelism involves much more than evangelists. It requires people to print Bibles, for instance, and people to distribute them, people to fund certain mission trips or projects, people to minister in various ways to the missionaries and the evangelists.
So while we're all called to witness and we're all responsible to some degree for seeing that the tasks of evangelism and mission are accomplished, we're not all meant to be missionaries or evangelists.